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From: Eric Angelini
To:
math-fun@mailman.xmission.com 26/02/2023
18:41
Hello
Math-Fun,
A few
thoughts and seq thingies.
---RUNS
of consecutive consonants:
ONE,
TWO, THREE, (repeat the pattern ad inf.)
---RUNS
of consecutive vowels:
TWO,
ONE, THREE, ONE, (repeat the pattern ad inf.)
---RUNS
of consecutive vowels _and_ consonants:
TWO,
ONE, THREE, ONE, (the same pattern as above!)
CUMULATIVE
sum of consonants used so far:
ONE,
_end
TWO,
FOUR, SIX, _end
THREE,
FIVE, EIGHT, TEN, FOURTEEN, EIGHTEEN, TWENTYFOUR, _end
THREE,
FIVE, EIGHT, ELEVEN, FIFTEEN, NINETEEN, TWENTYFIVE, _end
THREE,
FIVE, EIGHT, TWELVE, SIXTEEN, TWENTY, TWENTYSIX, THIRTY, ... (cont?)
THREE,
FIVE, EIGHT, TWELVE, SIXTEEN, TWENTY, TWENTYSIX, THIRTYONE, ... (cont?)
THREE,
FIVE, EIGHT, TWELVE, SIXTEEN, TWENTY, TWENTYSIX, THIRTYTWO, ... (cont?)
THREE,
FIVE, EIGHT, TWELVE, SIXTEEN, TWENTY, TWENTYSIX, THIRTYTHREE, ... (cont?)
THREE,
FIVE, EIGHT, TWELVE, SIXTEEN, TWENTYONE, TWENTYEIGHT, THIRTYFOUR, ... (cont?)
THREE,
FIVE, EIGHT, TWELVE, SIXTEEN, TWENTYTWO, _end
THREE,
FIVE, EIGHT, TWELVE, SIXTEEN, TWENTYTHREE, TWENTYNINE, ... (cont?)
CUMULATIVE
sums ALTERNATED (consonants, vowels):
---ONE
(con), FOUR (vow), FIVE (con), EIGHT (vow), TEN (con), TEN (vow), SIXTEEN
(con), SIXTEEN (vow), TWENTYSIX (con), TWENTYTWO (vow), __end
---ONE
(con), FOUR (vow), FIVE (con), EIGHT (vow), TEN (con), TEN (vow), SIXTEEN
(con), SIXTEEN (vow), TWENTYSIX (con), TWENTYTHREE (vow), THIRTYNINE
(con), THIRTYONE (vow), FORTYNINE (con), THIRTYNINE (vow), __end
____________________________________________________________
A recent private mail
to Gavin L. (Feb 26, 2023)
Hi
Gavin, how are you?
Would
you check, extend and submit with both our names the hereunder idea to the
OEIS?
If not,
let me know — and no hurry as usual! (I hope
this is not old hat…)
Best,
É.
DATA
1,2,21,12,3,11,4,44,5,6,22,221,111,7,77,71,1111,8,88,888,9,91,…
NAME
The
concatenation of the successive sizes of equal digits runs is the concatenation
of the sequence’s terms. This is the lexicographically earliest sequence of
distinct integers having this property.
COMMENT
As each
digit is a run’s size, no zero is present in the sequence.
EXAMPLE
The
first run has size 1 (the single 1 that starts the sequence);
The
second run has size 2 (two 2s);
The
third run has size 2 again (two 1s);
The
fourth run has size 1 (the single 2 of 12)
The
fifth run has size 1 again (the single 3)
The
sixth run has size 2 (the two 1s of 11);
The
seventh run has size 3 (the three consecutive 4s after 11); etc.
The
sequence is always extended with the smallest integer not yet used that doesn’t
lead to a contradiction.
Cf.
On Tue,
Feb 28, 2023, 12:39 Éric Angelini <eric.angelini@skynet.be> wrote a (stupid)
mail to Math-Fun:
> Hello
Math-Fun,
>… this
recent remark by Neil gave me an idea:
>> I
apply my favorite tools, DIFF and RUNS (...)
>… Here
is a draft for the OEIS:
NAME
Runs of
identical digits and first differences have the same concatenation.
DATA
1,2,24,40,3,333331,112,2222223,331,113,332,221,122,22222233,34,445,556,61,4,444444441,12,21,13,30,5,6,11,22,33,7,
COMMENT
R being
the size of a run, we want R>=1 and R<=9.
As we
want the sequence to be extended with the smallest positive integer not yet
present and not leading to a contradiction, we don’t accept the presence
of any digit 0 in a first difference (such a 0 would force a run of size
0, which is not accepted).
EXAMPLE
Runs 1 2 2 1 6 3
Seq 1,2, 24, 40, 3, 333331,112,
Diff 1
22 16 37
We see
that the concatenations of the terms of the first and the third row are
equal.
P.-S.
I am
almost 100% sure that the terms above went quickly wrong — but you get the idea.
Best,
É.
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_______________________________________________
Maximilian was quick to open my eyes:
Le 28
févr. 2023 à 19:39, M F Hasler <mhasler@dsi972.fr> a écrit :
But
then, why not
R = 1,1,1,1...
S = 1,2,3,4....
D
= 1,1,1,1...
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mailing list -- math-fun@mailman.xmission.com
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unsubscribe send an email to math-fun-leave@mailman.xmission.com
_______________________________________________
I've answered to Maximilian in private (and French):
Hello
Maximilian
Ta
remarque impeccable fut productive !
J’ai
compris ce que je voulais faire !-)
D’abord
la suite A où les RUNS sont les DIFF (sans
utiliser les concaténations);
puis la
suite B où les RUNS sont les DIFF aussi, mais en
concaténant (la suite que tu as invalidée).
Voici A
(modulo erreurs, pardon):
A=1,2,3,4,5,6,7,8,9,10,11,12,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,51…
Les RUNS
et DIFF de A sont identiques:
1 1 1 1
1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3
1 1…
Voici B
(beaucoup plus rock’n’roll — les RUNS et DIFF ne sont
pas identiques mais leurs concaténations oui):
B=1,2,3,4,5,6,7,8,9,11,12,15,14,13,24,23,22,21,20,19,18,17,48,37,26,25,36,35,34,33,32,31,30,29,28,27,16,127,114,103,92,81,70,59,58,47,46,45,44,42,…
RUNS B
(11x1) -
3 - (11x1) - 3 - (23x1) - 3 - (16x1) - 2 - (20x1) - 3…
DIFF B
(11x1) -
3 - (11x1) - 31 - 11 - 11 - 1 - 11 - (9x1) - 11- 111 - 13 - (5x11) - 1 - 11 - 1
- 1 - 1 - 2…
March 1st, 2023, from Eric Angelini to Math-Fun
Split in two parts and multiply them
Hello Math-Fun,
Take the integer 1133
We split 1133 into 1 and 133 for instance
(inserting a star between two digits).
We then make 1*133 = 133
We iterate until we get a single digit.
Question:
Is there an integer that can reach any of the 10 digits?
With 1133 we can reach 0, 4, 7, 8 or 9:
1133_11*33_36*3_10*8_8*0_0
1133_11*33_3*63_1*89_8*9_7*2_1*4_4
1133_113*3_3*39_1*17_1*7_7
1133_113*3_3*39_11*7_7*7_4*9_3*6_1*8_8
1133_1*133_1*33_3*3_9
P.-S.
As we don't want substrings with leading zeros, we don’t insert a star before a zero (except, if needed, before the integer 0 itself). Sorry if this is old hat.
Best,
É.
Update, March 4th, 2023
I love this puzzle – but I guess I am the only one!
Here is 389, (apparently) the first integer N going down the "river delta" from N to every single even digits:
If we split 389 into [3*89] we get 267
We then split 267 into [2*67] and get 134:
134 split into [1*34] gives 34 and [3*4] gives 12, which ends in [1*2] = 2;
134 split into [13*4] gives 52 and [5*2] gives 10, which ends in [1*0] = 0;
But we can also split 267 into [26*7] and get 182:
182 split into [1*82] gives 82 and [8*2] gives 16, which ends in [1*6] = 6;
182 split into [18*2] gives 36 and [3*6] gives 18, which ends in [1*8] = 8;
We could have split 389 into [38*9] and gotten 342:
We then split 342 into [3*42] and get 126:
126 split into [1*26] gives 26, and [2*6] gives 12, which ends in [1*2] = 2;
126 split into [12*6] gives 72, and [7*2] gives 14, which ends in [1*4] = 4.
We have our 5 (yellow) distinct even single-digit ends.
Who will find:
— the 1st integer that could end in 5 distinct odd digits?
— the 1st integer that could end in 10 distinct digits? (as mentionned above to the Math-Fun mailing list, I'm not sure this is possible)
— the 1st integer > 9 that produces its own digits, one after the other, in the right order, and nothing more (such integers, if they exist, could be called self-delta numbers)?
For the last challenge, we have to fix an "operations order"; this is easy:
a) split the integer a1a2a3a4a5a6a7 … an into [a1*a2a3a4a5a6a7 … an] and compute the result R; R will look like b1b2 b3 b4 b5 b6 b7 … bn;
b) go back to (a) with R until a single digit is reached and note it.
c) from where you are now, go up in the delta to the first unexplored branch, descend it and compute;
d) note the end digit of the second branch and go back to (c) until the whole delta is explored.
I've used the table below to compute the table below (!) Note that 12 and 21 end in the same branch of the delta.
Integers 0 to 9 end in themselves, then:
10 = 1*0 = 0
11 = 1*1 = 1
12 = 1*2 = 2
...
19 = 1*9 = 9
20 = 0
22 = 4
23 = 6
24 = 8
25 = 10 = 0
26 = 12 = 2
27 = 14 = 4
28 = 16 = 6
29 = 18 = 8
30 = 0
33 = 9
34 = 12 = 2
35 = 15 = 5
36 = 18 = 8
37 = 21 = 2
38 = 24 = 8
39 = 27 = 14 = 4
40 = 0
44 = 16 = 6
45 = 20 = 0
46 = 24 = 8
47 = 28 = 16 = 6
48 = 32 = 6
49 = 36 = 18 = 8
50 = 0
55 = 25 = 10 = 0
56 = 30 = 0
57 = 35 = 15 = 5
58 = 40 = 0
59 = 45 = 20 = 0
60 = 0
66 = 12 = 2
67 = 42 = 8
68 = 48 = 32 = 6
69 = 54 = 20 = 0
70 = 0
77 = 49 = 36 = 18 = 8
78 = 56 = 30 = 0
79 = 63 = 18 = 8
80 = 0
88 = 64 = 24 = 8
89 = 72 = 14 = 4
90 = 0
99 = 81 = 8
100 = 0
101 = 0
... = 0
110 = 0
111 = first split [1*11] = 11, then [1*1] = 1
111 = second split [11*1] = 11, then [1*1] = 1
112 = first split [1*12] = 12, then [1*2] = 2
112 = second split [11*2] = 22, then [2*2] = 4
113 = first split [1*13] = 13, then [1*3] = 3
113 = second split [11*3] = 33, then [3*3] = 9
114 = first split [1*14] = 14, then [1*4] = 4
114 = second split [11*4] = 44, then [4*4] = 16 = 6
115 = 15 = 5 and 115 = 55 = 25 = 10 = 0
116 = 16 = 6 and 116 = 66 = 36 = 18 = 8
117 = 17 = 7 and 117 = 77 = 49 = 36 = 18 = 8
118 = 18 = 8 and 118 = 88 = 64 = 24 = 8
119 = 9, 8
120 = 0, 0
121 = 2, 2
122 = 8, 4
123 = 8, 6
124 = 8, 6
125 = 0, 0
126 = 2, 4
127 = 4, 6
128 = 6, 0
129 = 8, 0
130 = 0, 0
131 = 3, 3
132 = 6, 2
133 = 9, 4
134 = 2, 0
135 = 5, 0
136 = 8, 0
137 = 2, 9
138 = 8, 0
139 = 4, 7, 8 (as 13*9 = 117 and 117 ends in 7 or 8, see above)
140 = 0, 0
141 = 4, 4
142 = 8, 6
143 = 2, 8
144 = 6, 0
145 = 0, 0
146 = 8, 6
147 = 6, 4
148 = 6, 2, 4 (as 14*8 = 112 and 112 ends in 2 or 4, see above)
149 = 8, 2, 4 (as 14*9 = 126 and 126 ends in 2 or 4, see above)
150 = 0, 0
151 = 5, 5
152 = 0, 0
153 = 5, 0
154 = 0, 0
155 = 0, 5
156 = 0, 0
157 = 5, 0
158 = 0, 0
159 = 0, 5, 0
160 = etc.
161 =
162 =
163 =
164 =
165 =
166 =
167 =
168 =
169 =
170 =
171 =
172 =
173 =
174 =
175 =
176 =
177 =
178 =
179 =
180 =
181 =
182 =
183 =
184 =
185 =
186 =
187 =
188 =
189 =
190 =
191 =
192 =
193 =
194 =
195 =
196 =
197 =
198 =
199 =
200 =
All ears come from here.
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